Keshawn_lu's Blog

吴恩达深度学习 C1_W3_Assignment

字数统计: 1.8k阅读时长: 10 min
2021/09/16 Share

吴恩达深度学习 C1_W3_Assignment

任务:实现单隐藏二类分类神经网络

Part0:库的准备

1
2
3
4
5
6
7
8
9
10
11
12
# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases import *
import sklearn
import sklearn.datasets
import sklearn.linear_model
from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets

%matplotlib inline

np.random.seed(1) # set a seed so that the results are consistent

Part1:准备数据

导入数据并观察

1
2
3
4
X, Y = load_planar_dataset()

# Visualize the data:
plt.scatter(X[0, :], X[1, :], c=Y[0, :], s=40, cmap=plt.cm.Spectral);

https://pic.imgdb.cn/item/61304dd844eaada73938973f.png

Exercise1:有多少训练样本,X和Y的维度又是怎么样的?

1
2
3
4
5
6
7
8
9
### START CODE HERE ### (≈ 3 lines of code)
m = X.shape[1]
shape_X = X.shape
shape_Y = Y.shape
### END CODE HERE ###

print ('The shape of X is: ' + str(shape_X))
print ('The shape of Y is: ' + str(shape_Y))
print ('I have m = %d training examples!' % (m))
1
2
3
The shape of X is: (2, 400)
The shape of Y is: (1, 400)
I have m = 400 training examples!

Part2:使用简单的逻辑回归来观察分类效果

1
2
3
4
5
6
7
8
9
10
11
12
# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y[0, :].T);

# Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x), X, Y[0, :])
plt.title("Logistic Regression")

# Print accuracy
LR_predictions = clf.predict(X.T)
print ('Accuracy of logistic regression: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +
'% ' + "(percentage of correctly labelled datapoints)")
1
Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)

https://pic.imgdb.cn/item/613051a044eaada7393d8e1d.png

由于数据集并不是线性可分的,所以效果并不是很好,接下来我们试试用神经网络进行分类。

Part3:神经网络模型

模型图示如下:

https://pic.imgdb.cn/item/6130530d44eaada7393f96a2.png

数学上表示:

成本函数如下:

构建神经网络的一般方法为:

  • 定义神经网络结构
  • 初始化模型参数
  • 循环(正向传播、计算损失、反向传播获得梯度、更新参数(梯度下降))

Exercise2:定义神经网络结构,定义参数

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
def layer_sizes(X, Y):
"""
Arguments:
X -- input dataset of shape (input size, number of examples)
Y -- labels of shape (output size, number of examples)

Returns:
n_x -- the size of the input layer
n_h -- the size of the hidden layer
n_y -- the size of the output layer
"""
### START CODE HERE ### (≈ 3 lines of code)
n_x = X.shape[0]
n_h = 4
n_y = Y.shape[0]
### END CODE HERE ###
return (n_x, n_h, n_y)
1
2
3
4
5
X_assess, Y_assess = layer_sizes_test_case()
(n_x, n_h, n_y) = layer_sizes(X_assess, Y_assess)
print("The size of the input layer is: n_x = " + str(n_x))
print("The size of the hidden layer is: n_h = " + str(n_h))
print("The size of the output layer is: n_y = " + str(n_y))
1
2
3
The size of the input layer is: n_x = 5
The size of the hidden layer is: n_h = 4
The size of the output layer is: n_y = 2

Exercise3:初始化模型参数

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
def initialize_parameters(n_x, n_h, n_y):
"""
Argument:
n_x -- size of the input layer
n_h -- size of the hidden layer
n_y -- size of the output layer

Returns:
params -- python dictionary containing your parameters:
W1 -- weight matrix of shape (n_h, n_x)
b1 -- bias vector of shape (n_h, 1)
W2 -- weight matrix of shape (n_y, n_h)
b2 -- bias vector of shape (n_y, 1)
"""

np.random.seed(2)

### START CODE HERE ### (≈ 4 lines of code)

W1 = np.random.randn(n_h, n_x) * 0.01
b1 = np.zeros((n_h, 1))

W2 = np.random.randn(n_y, n_h) * 0.01
b2 = np.zeros((n_y, 1))

### END CODE HERE ###

assert (W1.shape == (n_h, n_x))
assert (b1.shape == (n_h, 1))
assert (W2.shape == (n_y, n_h))
assert (b2.shape == (n_y, 1))

parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}

return parameters

Exercise4:实现前向传播

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
def forward_propagation(X, parameters):
"""
Argument:
X -- input data of size (n_x, m)
parameters -- python dictionary containing your parameters (output of initialization function)

Returns:
A2 -- The sigmoid output of the second activation
cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"
"""
# Retrieve each parameter from the dictionary "parameters"
### START CODE HERE ### (≈ 4 lines of code)

W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

### END CODE HERE ###

# Implement Forward Propagation to calculate A2 (probabilities)
### START CODE HERE ### (≈ 4 lines of code)

Z1 = np.dot(W1, X) + b1
A1 = np.tanh(Z1)
Z2 = np.dot(W2, A1) + b2
A2 = sigmoid(Z2)

### END CODE HERE ###

assert(A2.shape == (1, X.shape[1]))

cache = {"Z1": Z1,
"A1": A1,
"Z2": Z2,
"A2": A2}

return A2, cache

Exercise5:计算成本函数

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
# GRADED FUNCTION: compute_cost

def compute_cost(A2, Y, parameters):
"""
Computes the cross-entropy cost given in equation (13)

Arguments:
A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
Y -- "true" labels vector of shape (1, number of examples)
parameters -- python dictionary containing your parameters W1, b1, W2 and b2

Returns:
cost -- cross-entropy cost given equation (13)
"""

m = Y.shape[1] # number of example

# Compute the cross-entropy cost
### START CODE HERE ### (≈ 2 lines of code)
logprobs = np.multiply(np.log(A2), Y) + np.multiply((1 - Y), np.log(1 - A2))
cost = -1 / m * np.sum(logprobs)

### END CODE HERE ###

cost = np.squeeze(cost) # 把shape中为1的维度去掉
# E.g., turns [[17]] into 17
assert(isinstance(cost, float))

return cost

Exercise6:实现反向传播

https://pic.imgdb.cn/item/613057fd44eaada739468907.png

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
def backward_propagation(parameters, cache, X, Y):
"""
Implement the backward propagation using the instructions above.

Arguments:
parameters -- python dictionary containing our parameters
cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".
X -- input data of shape (2, number of examples)
Y -- "true" labels vector of shape (1, number of examples)

Returns:
grads -- python dictionary containing your gradients with respect to different parameters
"""
m = X.shape[1]

# First, retrieve W1 and W2 from the dictionary "parameters".
### START CODE HERE ### (≈ 2 lines of code)

W1 = parameters["W1"]
W2 = parameters["W2"]

### END CODE HERE ###

# Retrieve also A1 and A2 from dictionary "cache".
### START CODE HERE ### (≈ 2 lines of code)

A1 = cache["A1"]
A2 = cache["A2"]

Z1 = cache["Z1"]

### END CODE HERE ###

# Backward propagation: calculate dW1, db1, dW2, db2.
### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above)

dZ2 = A2 - Y
dW2 = 1 / m * np.dot(dZ2, A1.T)
db2 = 1 / m * np.sum(dZ2, axis=1, keepdims=True)

dZ1 = np.dot(W2.T, dZ2) * (1 - np.power(A1, 2))
dW1 = 1 / m * np.dot(dZ1, X.T)
db1 = 1 / m * np.sum(dZ1, axis=1, keepdims=True)

### END CODE HERE ###

grads = {"dW1": dW1,
"db1": db1,
"dW2": dW2,
"db2": db2}

return grads

Exercise7:使用梯度下降来更新参数

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
def update_parameters(parameters, grads, learning_rate = 1.2):
"""
Updates parameters using the gradient descent update rule given above

Arguments:
parameters -- python dictionary containing your parameters
grads -- python dictionary containing your gradients

Returns:
parameters -- python dictionary containing your updated parameters
"""
# Retrieve each parameter from the dictionary "parameters"
### START CODE HERE ### (≈ 4 lines of code)

W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

### END CODE HERE ###

# Retrieve each gradient from the dictionary "grads"
### START CODE HERE ### (≈ 4 lines of code)

dW1 = grads["dW1"]
db1 = grads["db1"]
dW2 = grads["dW2"]
db2 = grads["db2"]

## END CODE HERE ###

# Update rule for each parameter
### START CODE HERE ### (≈ 4 lines of code)

W1 = W1 - learning_rate * dW1
b1 = b1 - learning_rate * db1
W2 = W2 - learning_rate * dW2
b2 = b2 - learning_rate * db2

### END CODE HERE ###

parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}

return parameters

Exercise8:总结之前的函数,创建神经网络模型

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):
"""
Arguments:
X -- dataset of shape (2, number of examples)
Y -- labels of shape (1, number of examples)
n_h -- size of the hidden layer
num_iterations -- Number of iterations in gradient descent loop
print_cost -- if True, print the cost every 1000 iterations

Returns:
parameters -- parameters learnt by the model. They can then be used to predict.
"""

np.random.seed(3)
n_x = layer_sizes(X, Y)[0]
n_y = layer_sizes(X, Y)[2]

# Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".
### START CODE HERE ### (≈ 5 lines of code)

parameters = initialize_parameters(n_x, n_h, n_y)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]

### END CODE HERE ###

# Loop (gradient descent)

for i in range(0, num_iterations):

### START CODE HERE ### (≈ 4 lines of code)
# Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
A2, cache = forward_propagation(X, parameters)

# Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
cost = compute_cost(A2, Y, parameters)

# Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
grads = backward_propagation(parameters, cache, X, Y)

# Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
parameters = update_parameters(parameters, grads, learning_rate=1.2)

### END CODE HERE ###

# Print the cost every 1000 iterations
if print_cost and i % 1000 == 0:
print ("Cost after iteration %i: %f" %(i, cost))

return parameters

Exercise9:实现预测函数

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
def predict(parameters, X):
"""
Using the learned parameters, predicts a class for each example in X

Arguments:
parameters -- python dictionary containing your parameters
X -- input data of size (n_x, m)

Returns
predictions -- vector of predictions of our model (red: 0 / blue: 1)
"""

### START CODE HERE ### (≈ 2 lines of code)

A2, cache = forward_propagation(X, parameters)
predictions = (A2 > 0.5)

### END CODE HERE ###

return predictions

运行模型,观察结果:

1
2
3
4
5
6
# Build a model with a n_h-dimensional hidden layer
parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)

# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y[0, :])
plt.title("Decision Boundary for hidden layer size " + str(4))

https://pic.imgdb.cn/item/613059ea44eaada739498ab3.png

我们来观察一下准确率:

1
2
3
# Print accuracy
predictions = predict(parameters, X)
print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')
1
Accuracy: 90%

可以看出,与逻辑回归相比,准确率提高了很多。

CATALOG
  1. 1. 吴恩达深度学习 C1_W3_Assignment
    1. 1.1. 任务:实现单隐藏二类分类神经网络
    2. 1.2. Part0:库的准备
    3. 1.3. Part1:准备数据
      1. 1.3.1. Exercise1:有多少训练样本,X和Y的维度又是怎么样的?
    4. 1.4. Part2:使用简单的逻辑回归来观察分类效果
    5. 1.5. Part3:神经网络模型
      1. 1.5.1. Exercise2:定义神经网络结构,定义参数
      2. 1.5.2. Exercise3:初始化模型参数
      3. 1.5.3. Exercise4:实现前向传播
      4. 1.5.4. Exercise5:计算成本函数
      5. 1.5.5. Exercise6:实现反向传播
      6. 1.5.6. Exercise7:使用梯度下降来更新参数
      7. 1.5.7. Exercise8:总结之前的函数,创建神经网络模型
      8. 1.5.8. Exercise9:实现预测函数